关键词: 球谐级数, 分数阶导数, 拉普拉斯方程, 磁梯度测量

Abstract:  Adjusting the derivative order of fractional-order magnetic gradient (FMG) can tune the resolution of magnetic materials at different depths. However, direct FMG measurement methods remain lacking. Therefore, this article focuses on the research of fractional-order magnetic gradient measurement method. This article uses the external source terms of the Laplace equation solved in a spherical coordinate system as the model of the magnetic field inside the sphere, calculates the fractional-order derivatives in the corresponding directions of the model, and achieves FMG measurement. We evaluate sensitivity to measurement position error and field-measurement error through simulation, showing that the normalized root mean square error  increases monotonically with the derivative order. An experiment is conducted to verify the measurement method, and the relative error between the experimental results and the theoretical values does not exceed 6.00%, verifying the correctness of the measurement method.

Key words: spherical harmonics, fractional-order derivative, Laplace equation, magnetic gradient measurement